Structure of some Semihyperrings of Linear Transformations

For a semigroup S, the semigroup S0 is defined to be S if S has a zero and S contains more than one element, otherwise, let S0 be S with a zero 0 adjoined. We say that a semigroup S admits the structure of a semihyperring with zero if there exists a hyperoperation + on S0 such that (S0,+, •) is a semihyperring with zero where • is the operation on S0. Semigroups admitting the structure of an additively commutative [AC] semiring with zero are defined analogously. Let V be a vector space over a division ring R, W a subspace of V , LR(V,W) the semigroup under the composition of all linear transformations _ : V → W, and PLR(V,W) the partial linear transformation semigroup from V into W, the semigroup under the composition of all linear transformations from a subspace of V into W. If subsemigroups of LR(V,W) contain a zero, we determine when they admit the structure of a semihyperring with zero. Otherwise, we characterize when they admit the structure of an AC semiring with zero. Moreover, necessary conditions for PLR(V,W) to admit the structure of an AC semiring with zero are given.